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Properties

Label 439824.dz
Number of curves $4$
Conductor $439824$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("dz1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 439824.dz have rank \(1\).

Complex multiplication

The elliptic curves in class 439824.dz do not have complex multiplication.

Modular form 439824.2.a.dz

Copy content sage:E.q_eigenform(10)
 
\(q + q^{3} - 2 q^{5} + q^{9} + q^{11} + 2 q^{13} - 2 q^{15} - q^{17} - 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content sage:E.isogeny_class().matrix()
 

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

Copy content sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.

Elliptic curves in class 439824.dz

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
439824.dz1 439824dz4 \([0, 1, 0, -1100929664, -14060436701004]\) \(306234591284035366263793/1727485056\) \(832458298791297024\) \([2]\) \(74317824\) \(3.5084\)  
439824.dz2 439824dz2 \([0, 1, 0, -68809344, -219703209804]\) \(74768347616680342513/5615307472896\) \(2705962225167325200384\) \([2, 2]\) \(37158912\) \(3.1618\)  
439824.dz3 439824dz3 \([0, 1, 0, -64293504, -249780510540]\) \(-60992553706117024753/20624795251201152\) \(-9938888853539079502430208\) \([2]\) \(74317824\) \(3.5084\)  
439824.dz4 439824dz1 \([0, 1, 0, -4584064, -2955734860]\) \(22106889268753393/4969545596928\) \(2394775838445495386112\) \([2]\) \(18579456\) \(2.8152\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 439824.dz1.